Simultaneous Indirect Inference, Impulse Responses, and ARMA models

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1 Simultaneous Indirect Inference, Impulse Responses, and ARMA models Lynda Khalaf y and Beatriz Peraza López z August 2015 Abstract Indirect Inference based methods are proposed to derive Identi cation Robust con dence sets. The primary focus concerns ARMA models and impulse responses, in which case simultaneous con dence bands are derived. Auxiliary estimating functions include two-sided forward and backward looking regressions with leads and lags, long-autoregressions and empirical autocorrelations. Resulting objective functions are treated as test statistics that are inverted rather than optimized, using the Monte Carlo test method. Simulation studies illustrate accurate size and good power. Overall, results underscore the usefulness of proposed methods regardless of proximity to boundaries. Keywords: Con dence Set, Indirect Inference, ARMA, Impulse-Response, Monte Carlo test, Simultaneous Inference.. This work was supported by the Social Sciences and Humanities Research Council of Canada and the Fonds de recherche sur la société et la culture (Québec). y Department of Economics, Carleton University. Mailing address: Department of Economics, Room C-870 Loeb Building,Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6 Canada. Tel (613) ; FAX: (613) Lynda_Khalaf@carleton.ca. z Department of Economics, Carleton University. Mailing address: Department of Economics, Room C-870 Loeb Building, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6 Canada. Tel (613) ; FAX: (613) beatrizperazalopez@cmail.carleton.ca 1

2 1 Introduction Indirect inference refers to resampling-based statistical methods that use simulation to calculate estimating functions including likelihoods, scores or moments. The idea of replacing complicated or intractable statistical functions by computer simulations motivated initial applications and underlying theory; see Smith (1993), Gouriéroux, Monfort and Renault (1993), Gouriéroux and Monfort (1996) and Gallant and Tauchen (1996). Since then, available evidence suggests that indirect inference often works better than traditional methods in nite samples for bias or misspeci cation corrections. 1 This paper considers a two-stage framework that can yield identi cation robust con dence sets via Indirect Inference. The primary focus concerns inference in Autoregressive Moving Average (ARMA) models including impulse responses, in which case we derive simultaneous con dence bands, as in Jordá (2009); see also Jordá and Marcellino (2010) and Jordá, Knuppel and Marcellino (2013) on simultaneous path forecasts. Despite the simplicity of ARMA models in time series, identi cation and boundary issues raise enduring complications for estimation and inference. Autoregression (AR) unit roots cause the so-called non-uniform convergence problem: methods that require stationarity break down at or close to the unit boundary, whereas local-to-unity motivated methods perform poorly away from unity; see Hansen (1999), Mikusheva (2007), Andrews and Guggenberger (2010), and more recently, Gorodnichenko, Mikusheva and Ng (2012), Mikusheva (2012, 2014) and Phillips (2014). Invertibility of Moving Average (MA) components causes the so-called pile up e ect: because of identi cation failure, estimating functions including likelihoods can have a local maximum at the invertibility boundary even when the true process is invertible; see Sargan and Bhargava (1983), Davis and Dunsmuir (1996), Gospodinov (2002), and more recently, Davis and Song (2011), and 1 See, for example, Dridi, Renault and Guay (2007), Gouriéroux, Phillips and Yu (2010), Li (2010), Dominicy and Veredas (2013), Fuleky and Zivot (2014) and Calvet and Czellar (2015). 2

3 Gospodinov and Ng (2015). We propose con dence sets robust to both problems via Indirect Inference and the following auxiliary estimates: (i) two-sided forward and backward looking regressions with leads and lags, (ii) long-autoregressions (AR), and (iii) empirical autocorrelations. Methods with leads and lags generally aim to deal with feedback e ects; see Stock and Watson (1993), Dufour and Torrès (2000). AR-based approximations have long been popular for ARMA models; see Hannan and Rissanen (1982), Galbraith and Zinde-Walsh (1994, 1997), Ghysels, Khalaf and Vodounou (2002), Guay and Scaillet (2003), Gospodinov and Ng (2015), Dufour and Pelletier (2014). Empirical autocorrelations and related measures including cumulants and local projections have recently been re-introduced from a minimum distance perspective; see e.g. Jordá and Kozicki (2011), Gorodnichenko, Mikusheva and Ng (2012), Gospodinov and Ng (2015) and the references therein. The indirect inference framework we propose uni es all the above for con dence interval purposes. While the ARMA case concretizes concepts, our method applies more generally, as follows. Indirect inference relies on a binding function, denoted thereafter = L(); that links a parameter of interest to an auxiliary parameter for which a simple estimator is available. Denote the latter ^ = ^(X) where X refers to available data and ^(:) is the estimating function. L() need not be known but admits a simulation-based approximate. Denote the latter (). Conformably, a minimum-distance estimator is derived that matches ^ to (). To set focus, consider a Wald-type distance metric of the form S(X; ) = (^ ()) 0 (^ ()). The standard Indirect Inference estimator corresponds to ^ = ARGMIN (S(X; )). Associated con dence sets and hypotheses tests are typically justi ed using standard asymptotic arguments. For example, regularity conditions are derived to ensure consistency and e.g. asymptotic normality of ^, leading to Wald-type con dence intervals. Such intervals usually require identifying, which may be an issue in many models that necessitate Indirect Inference. We revisit Indirect Inference from an 3

4 alternative outlook, to demonstrate its usefulness when identi cation of raises di culties, and when further parameters of interest are m non-standard transformations of, as with impulse responses, which we denote g i (), i = 1; : : : ; m. In contrast to the above described extremum estimation strategy, we treat S(X; ) as a test statistic which we invert rather than minimize. Inverting a test involves collecting the parameter values that are not rejected via this test at a given level of signi cance to construct a con dence set. We propose to automate the latter into Indirect Inference, along these lines. Assume that () can be derived by simulating the true process once is set to a given value, say 0. Then a simulation-based p-value can be derived for S(X; 0 ) = (^ ( 0 )) 0 (^ ( 0 )) using this same 0 and a supplementary simulation round conditional on ( 0 ). Denote the p-value thus derived as ^p ( 0 ). Simulations are thus incorporated in two stages, rst to calibrate ( 0 ) in S(X; 0 ) and next to compute its p-value. An -level joint con dence set can thus be assembled by numerically collecting the 0 values for which ^p ( 0 ) >. Exactness and identi cation robustness are achieved by computing ^p ( 0 ) via the MC test method of Dufour (2006). This method achieves size control when the null distribution of the statistic, even if not computable analytically, does not depend on unknown or nuisance parameters. In this regard, our work relates to Dufour and Valery (2006, 2009) in stochastic volatility models, and Dufour and Kurz-Kim (2010) and Beaulieu, Dufour and Khalaf (2007, 2013, 2014) in models with fat-tailed fundamentals. Many applications of Indirect Inference satisfy this requirement nearly by de nition, as () should be computable once is set. In our ARMA application, a nuisance-parameter-free distribution results from scale-invariance of OLS in AR settings, and from the standardization inherent to empirical autocorrelations. 2 For inference on g i (), i = 1; : : : ; m, the join con dence set 2 Scale invariance is not an unduly restrictive requirement particularly in univariate models. For example, Gouriéroux, Phillips and Yu (2010) consider a unit variance for Indirect Inference on the lag coe cient in dynamic Panels; Beaulieu, Dufour and Khalaf (2014) standardize scale exactly to one for MC tests on 4

5 is projected, which entails maximizing and minimizing each function over the non-rejected values of. The con dence intervals obtained are thus simultaneous by construction. We conduct a simulation study on ARMA and MA processes and compare our method with MLE. Simple objective functions as in Galbraith and Zinde-Walsh (1994, 1997) as well as over-identi ed criteria are considered, in addition to various lag structures. Results can be summarized as follows. In contrast to MLE, our method eradicates pile up problems even with very small samples, with no more than 50 observations. The signi cance level is a ected neither by the at segments in the objective function, nor by how close the estimated parameters are to one. MLE is oversized for both individual parameter and joint tests. None of the estimation functions steadily outperforms the rest for all sample sizes and pairs tested. Leads improve power for highly persistent processes. Overidenti cation yields important power bene ts. On balance, autocorrelation-based methods require a larger sample to catch up with AR-based counterparts. Inverting the asymptotic Wald test based on MLE estimates is almost infeasible at corner solutions with parameters close to the unit boundary, since in many instances the variance-covariance matrix is not invertible. This result is expected and provides support to our research. Finally, we show that MLE-based impulse-responses can severely underestimate the e ect of shocks. The paper is organized as follows. In section 2, we introduce our framework. In section 3, we discuss inference by test inversion. The ARMA special case is presented in section 4. Simulation results are presented in section 5. We conclude in section 6. tail parameters of Stable distributions. 5

6 2 General Framework Consider the general model (X ; fp (#) : # 2 g) (1) where X is the sample space, X = (X 1 ; :::; X t ; :::; X T ) 0 (2) represents a sample of size T, P (#) is a probability distribution over X indexed by # = ( 0 ; 0 ) 0 ; 2 ; 2 (3) and is the q-dimensional parameter of interest. is a nuisance parameter one aims to partial out. Pseudo-data can be simulated from (1) and an auxiliary parameter denoted, of dimension p q, can be de ned so that it is linked to via an exact or limiting binding function = L(#). In general, L(:) need not have a known analytical form. Furthermore, a simple statistic is available to estimate given X. Denote by ^ = ^(X) the considered data-based estimate of where ^(:) is the estimating function. Its simulation-based counterpart given a speci c value of # can be derived, for example, by: (i) creating H simulated paths x h (#) = (x h;1 (#); :::; x h;t (#):::; x h;t (#)) 0 ; h = 1; :::; H (4) each of size T from (1) given #; (ii) applying the same estimation method that was used 6

7 to derive ^ to each simulated path, leading to a series of estimators ^ h (#) = ^(x h (#)); h = 1; :::; H (5) and then (iii) computing their average: (#) = 1 H HX ^ h (#): (6) h=1 The distance between ^ and (#) forms the basis of Indirect Inference. A common special case is the Wald-type distance measure (^ (#)) 0 ^(^ (#)) where ^ is a positive de nite weighting matrix. Gouriéroux et al. (1993) suggest ^ = I p where I p denotes a p-dimensional identity matrix because the loss in e ciency with respect to an optimal estimator can be disregarded. This practice is also suggested, more recently, by Gouriéroux, Phillips and Yu (2010). 2.1 Nuisance parameters Focusing on as the parameter of interest requires a strategy to account for. In some settings, can be easily estimated given. Alternatively, an auxiliary estimate or objective function that is exactly invariant to can be considered. We emphasize the latter case in this paper, de ned via the following assumptions that characterize the model and considered estimating functions. Assumption 1. In the context of model (1); an estimating function ^(:) is invariant to, if ^(X ) = ^ X y (7) where X and X y are samples from (1) conformable with probability distributions P (( 0 ; 0 ) 0 ) 7

8 and P ( 0 ; 0 y )0 ; respectively, and 6= y are two di erent values of : This assumption ensures that ^(x h (( 0 ; 0 ) 0 )) = ^(x h (( 0 ; 0 y) 0 )); h = 1; :::; H which yields an invariant objective function for the transformation that maps X into X y. Indeed, applying Assumption 1 to the calibration in (6), we have: (( 0 ; 0 ) 0 ) = 1 H = 1 H HX ^ h (( 0 ; 0 ) 0 ) (8) h=1 HX ^ h (( 0 ; 0 y) 0 ) = (( 0 ; 0 y) 0 ): (9) h=1 It follows that can be set to any value in for simulation purposes with no bearing on estimation nor inference via objective functions of the form S(X; #) = (^ (#)) 0 (^ (#)): (10) In line with the above cited literature, (10) uses an identity weighting matrix. Other choices are possible as long as S(X; #) remains invariant to, or that the distribution of S(X; #) does not depend on under the null hypothesis that xes. The latter pivotality assumption does not require numerical invariance to throughout the considered parameter space. Instead, a - possibly restricted - estimate of may be available that evacuates the parameter only under the null. While Assumption 1 su ces for the ARMA models we consider below, we formalize the alternative pivotality assumption for completion as follows. 8

9 Assumption 2. In the context of model (1) and the null hypothesis H 0 ( 0 ) : = 0 ; 0 known an objective function S(X; ( 0 ; ^ 0 ) 0 )) where ^ is a conformable estimate of, is invariant to if the null distribution of S(X; ( 0 0; ^ 0 ) 0 )) does not depend on : A common special case includes scale invariance which implies robustness to multiplying the data by any positive scalar a, so Assumptions 1 yields: x h (( 0 ; a) 0 ) = ax h (( 0 ; 1) 0 ) ^(x h (( 0 ; 1) 0 )) = ^(x h (( 0 ; a) 0 )); ^(ax)) = ^ (X) : Its multivariate counterpart involves multiplying the data by an invertible matrix. For illustrative nite sample cases, see Beaulieu, Dufour and Khalaf (2007, 2013) and Dufour, Khalaf and Beaulieu (2003, 2010). In these papers, the inverted tests for multivariate Student-t and mixtures of normals distributions require Assumption 2, and the estimate in question uses the sum-of-squares residuals matrix. To simplify notation in what follows, since is partialled-out, that is, will be set in simulations to any known value in it parameter space, we suppress it out in the notation for S(X; #), (#), ^ h (#) and x h (#), which we will rede ne as S(X; ), (), ^ h () and x h (). 9

10 3 Inference via test inversion Standard applications of Indirect Inference resort to minimizing (10) and formulating adequate regularity conditions so that the resulting estimator, denoted ^, is (typically) asymptotically normal with a covariance matrix that can be estimated consistently. To compare and contrast our proposed inference method with common practices and to de ne test inversion within a familiar setting, let us revisit the traditional -level con dence interval CI(g(); ) = fg(^) z ^1=2 =2 g g for a given scalar function g (), where ^ 1=2 g denotes the relevant standard error estimate computed (for example, via the delta-method) from the estimated asymptotic variance/covariance matrix of ^, and z =2 is two-tailed standard normal critical point. As is well known, CI(g(); ) is derived by solving, over g( 0 ), the following inequality jg(^) g( 0 )j=^ 1=2 g z =2 : (11) Said di erently, the solution of (11) inverts a t-type test of H g : g() = g( 0 ); 0 known (12) based on the statistic jg(^) g()j=^ 1=2 g. Ideally, P[g() 2 CI(g(); )] 1, at least as T! 1. However, identi cation failures and the above cited pile up e ects, root cancellations or stationarity concerns will cause intervals of this form to severely under-cover, even with large samples. In this paper, we depart from this approach and propose a con dence set for based on inverting a test 10

11 of H 0 ( 0 ) : = 0 ; 0 known (13) using (10) which we treat as a test statistic. Inverting a test of H 0 ( 0 ) at a given level consists in collecting, numerically or analytically, the 0 values that are not rejected using the considered test at the considered level. For example, given the right-tailed test statistic T ( 0 ) = S(X; 0 ) = (^ ( 0 )) 0 (^ ( 0 )) (14) and assuming (for the moment and for illustrative purposes) that an -level cut-o point T c is available, test inversion involves solving, over 0, the inequality T ( 0 ) < T c. The solution of this inequality is a parameter space subset, denoted CS(; ), that satis es P[ 2 CS(; )] 1 : (15) Since the nite sample distribution of (10) is intractable, and since there is no reason to expect that a cut-o point that is invariant to 0 can be found, a numerical solution is required. We propose to collect by numerical search, the 0 values that are not rejected using a MC p-value we will de ne in the next section, denoted ^p ( 0 ). In other words, we propose to search for the 0 values for which ^p ( 0 ) >. The output of such a search is a joint -level con dence region, which we denote CS(; ) as above. The set thus derived can be empty, which will provide a diagnostic test of the hypothesized model. Said di erently, an empty con dence set would occur when all values of are rejected at the considered level: in this case, the considered speci cation would be incompatible with available data. Our proposed variant of Indirect Inference thus preserves its speci cation-robustness attributes. If identi cation is weak, the set will be unbounded or di use, which will also provide useful 11

12 information regarding model t (or lack thereof). If a con dence interval for the individual components of or, more generally, for a given function g() is desired, the above de ned CS(; ) can be projected. This implies minimizing and maximizing g() over the values in CS(; ). The ARMA-based impulse-response special case we study below illustrates the usefulness of such projections. 3.1 Exact p-values We propose applying Dufour s (2006) MC method to compute the above ^p ( 0 ) empirical p-value. Given that the null distribution of the considered statistic is nuisance parameter free, the MC method controls type I error even if S(X; 0 ) is itself simulation-based, in nite samples. The method can be summarized as follows. We introduce a new layer of simulation, for each value of 0 : we draw L replications satisfying the null hypothesis from the model, regardless the choice for x l ( 0 ) = (x l;1 ( 0 ); :::; x l;t ( 0 ); :::; x l;t ( 0 )) 0 ; l = 1:::L: (16) These replications should be generated independently from the simulations underlying S(X; 0 ). Next, we apply ^(:) to each replication l leading to: ~ l ( 0 ) = ^(x l ( 0 )); l = 1; :::; L; (17) and compute a series of statistics similar to the one in (14), but now considering the vector of the estimated parameters ~ l ( 0 ) for each replication l S l (x l ( 0 )) = ( ~ l ( 0 ) ( 0 )) 0 ( ~ l ( 0 ) ( 0 )) (18) 12

13 instead of the vector obtained by applying the estimation method to the data. We are able to use the same vector of calibrated parameters ( 0 ) from (14) due to the exchangeability property of the MC test [Dufour (2006)]. Lastly, the empirical p-value associated with the null hypothesis is computed as: ^p ( 0 ) = G L + 1 L + 1 ; (19) where G L denotes the number of times the statistic S l (x l ( 0 )) is greater than or equal to the data based S(X; 0 ): 4 ARMA special case To set focus, we consider the zero mean Gaussian ARMA(1,1) process of size T with unknown parameters and, X t = X t 1 + e t + e t 1 ; t = 1:::T; (20) with e t = u t (21) where jj < 1 guarantees invertibility or an Autoregressive AR(1) representation, j j < 1 ensures stationarity, and the joint distribution of u 1 ; :::; u t ; :::; u T is known. For example, u t can be assumed independent and identically distributed with a standard normal distribution: u t i:i:d: N (0; 1): (22) We assume that X 0 = e 0 where e 0 is xed. Student-t errors are also considered in our Monte Carlo study; the generalized lambda family of distributions as considered in Gospodinov and Ng (2015) is another interesting parametrization. With regards to the above notation, 13

14 we maintain as the nuisance parameter and we focus on joint estimation of = (; ) 0 : In this context, for any positive scalar a, we have: Q(X) = minq(x) = minq(ax) = aminq p (X) (23) = 0 f; 0 b 0 ; f = f1 ; :::; fp 0 ; b = b1 ; :::; bp 0 (24) T P t=p+1 [X t f1 X t+1 ::: fp X t+p b1 X t 1 ::: bp X t p ] 2 ; (25) and ^ j = d Cov(X t ; X t j ) dv ar(x t ) Cov(aX = d t ; ax t j ) ; j = 1:::p: (26) dv ar(ax t ) These invariance properties guarantees that Indirect Inference on based on Q(:) or on a set of empirical autocorrelations ^ j does not require specifying. To simplify notation, we use the same number of leads and lags in Q(X) although clearly, invariance to does not require this restriction. Conformably, we consider three choices for our estimating function ^(:): (i) two -sided OLS estimation with forward and backward-looking components; (ii) OLS estimation of backward-looking or long-ar; and (iii) empirical autocorrelations. For clarity, and given any sample vector X = (X 1 ; :::; X T ) 0 of size T, we will refer to these choices as ^ fb (X) = ARGMIN Q(X); with = 0 f; 0 b 0 ^ b (X) = ARGMIN 0 Q(X), with 0 = (0; 0 b) 0 ; ^ c (X) = (^ 1 ; :::; ^ p ) 0 where Q(X) corresponds to (25) and ^ j is given by (26). In each case, to compute the simulation-based counterparts of each for any given value of, it su ces to create H 14

15 simulated paths x h () = (x h;1 ; :::; x h;t ; :::; x h;t ) 0, as x h;t () = x h;t 1 + u h;t + u h;t 1 ; t = 1:::T; h = 1:::H: (27) The auxiliary parameter estimator for each path using all above estimating functions will numerically coincide with its counterpart based on x h;t () = x h;t 1 + u h;t + u h;t 1 ; t = 1:::T; h = 1:::H; (28) for any positive. Population autocorrelations need not be derived by simulation; yet invariance is explicitly imposed this way for any p. For the MA(1) special case, we also use a simpli ed auxiliary parameter: we t a long-ar to observed and simulated data and retain the coe cient of the rst lag X t 1 as ^. We denote the underlying estimating function as (:). 4.1 Impulse-response con dence bands Impulse-response functions are the most empirically and policy relevant transformations of ARMA parameters. If the stationarity condition is satis ed, (20) admits an MA (1) approximation using lags operators L, X t = (1 L) 1 (1 + L)e t 1X = (1 (L))e t = s e t s ; t = 1:::T; (29) s=0 15

16 and the empirical impulse-response coe cients can be computed following the well-known iterative process: 0 = 1 (30) s = s 1 ( + ) for s 1: (31) These relations de ne a series of known functions of the vector = (; ), denoted g i (), i = 1; :::; m, that can be projected in turn. This implies minimizing and maximizing g i () over the values in CS(; ). Con dence intervals so obtained are simultaneous, in the following sense: let g i CS (; ) denote the image of CS (; ) by the function g i. Then (15) implies that P g i () 2 g i CS (; ) ; i = 1; : : : ; m 1 : (32) In this context, our method is related to the research by Jordá (2009), Jordá and Marcellino (2010) and Jordá, Knuppel and Marcellino (2013) on simultaneous con dence bands and applications to forecasting. The simulation study reported below includes an illustrative analysis of such projections emphasizing close to boundary values of and : 5 Simulation Study We conduct MC simulation experiments to asses the performance of our method for MA(1) and ARMA(1,1) processes. Our objective is to study power and size when the MA parameters are close to the non-invertibility region for both processes, and the AR parameter is close to the non-stationary region for the ARMA. Our experiments are designed as follows. We simulate Gaussian processes and processes where the errors are drawn from 16

17 a Student- t distribution with 5 degrees of freedom. Simulated samples with number of observations T = 50, 100 and 200, used here as pseudo-data, are drawn from the ARMA model in (11), setting = 1 in line with in Assumptions 1 and 2. These sample size choices correspond to common samples selected in MC studies as well as data used in empirical applications. Each pair of fm A; ARg parameters used to generate ARMA(1,1) processes is denoted in our experiments as f 0 ; 0 g. The outcomes from all our simulation studies are available upon request. Here we focus on the following set of pairs to illustrate our results: f 0 ; 0 g = ff0; 0:6g; f0:6; 0g; f0:99; 0gf0:5; 0:96g; f0:96; 0:5g; f0:99; 0:99gg. In particular, we set the AR parameter = 0 in (20) a priory to draw samples from MA(1) processes with the values f 0 g = f0:6; 0:99g. To nd the number of auxiliary parameter estimates providing the highest power, we studied power under various combinations of the initial settings of our model. We changed the number of parameters from 1 to 3 and 8 for the MA and from 6 to 8 and 12 for the ARMA, taking into consideration the recommendations from Ghysels, Khalaf and Vodounou (2002) and Galbraith and Zinde-Walsh (1994) for the long-ar estimation, and Gorodnichenko et al. (2012) for the estimation with autocorrelations in our experiments. Since we found that 8 provides the highest power for all the sample sizes, we suggest that for empirical practices, the total number of auxiliary parameters is set to 8 on each of the choices of the estimating function explained in section 2.2: (i) f = b = 4 in the OLS estimation with forward and backward-looking components, (ii) = 8 in the OLS estimation of the long-ar, and (iii) j = 8 in the estimation with empirical autocorrelations. In the case of the simpli ed auxiliary parameter estimation for the MA process, the long- AR is estimated as (ii), but only the rst coe cient is retained. 17

18 In all cases, we set the number of simulated paths to H = 3. We tested H = 1 and 3 as suggested by Ghysels, Khalaf and Vodounou (2002) for the MA, and also H = 5 to examine whether it could improve power for ARMA. Then, we selected H = 3, keeping the number of auxiliary parameters equal to 8, as the value which provides the highest power for all the sample sizes. The number of replications in the application of Dufour s (2006) MC method to compute the p-values is set to L = 199, and N = 1000 replications are considered for the MC simulations. Lastly, an identity weighting matrix is used to preserve the invariance to scale of the statistic associated to every choice of the estimating function. We report empirical rejections and compare our results versus the MLE individual parameter tests and joint tests (Wald-type) in terms of size, considering a level of signi cance = 0:05. Our main results are summarized along the following lines. The results of the experiments conducted for MA(1) processes are presented in Tables 1 and 2, and the studies related to the ARMA(1,1) process are shown in Tables 3 to 6. We use abbreviations for each estimating function: OLS-FBLC, OLS-long AR and AutoCorr correspond to the OLS estimation with forward and backward-looking components, the OLS estimation of the long-ar and the estimation with empirical autocorrelations, respectively. Size is controlled with our method for all sample sizes, choices of the estimating function, and for both Gaussian and Student- t errors, irrespective of how close the true parameters are to the unit root. The MLE shows size problems when the sample size is small T = 50, for both, individual parameter tests and joint tests. Size seems to improve for the MLE when the sample size is increased and the parameters are far from the unit root regions. However, when the parameters approach the unit root regions, the MLE size fails as the sample size increases, which con rms its asymptotic failure. This is observed in Table 2, when 0 = f0:99g for the MA, and in Table 6 with the almost unit root pair f 0 ; 0 g = 18

19 f0:99; 0:99g for the ARMA. The results obtained in term of power for processes with Student- t errors are in line with the ones obtained with Gaussian errors. This outcome con rms the robustness to fatter tails, since we purposely ignore the fact that the errors are drawn from the Studentt distribution when we apply our method. From the experiments for the MA processes, in Tables 1 and 2, we can identify that the pile up e ect starts for values of the parameter higher than 0.6 for nite samples. Although our method does not produce size distortions, it cannot discriminate between tested values higher than 0.85 when the true value is between 0.85 and 1. For small values of the parameter, the method with the simpli ed auxiliary parameter estimation inspired in Galbraith and Zinde-Walsh (1994, 1997) shows an increase in power with respect to the OLS estimation of the long-ar. However, as the value of the parameter increases and gets closer to the unit root region, the latter method shows higher power than the former. This result indicates that overidenti cation is useful to increase power when we have higher persistence in MA(1) processes. In general, the power of our method increases for the three estimating functions applied to the ARMA as the sample size increases, but none of them steadily shows the highest power for all sample sizes and pairs tested. In Table 3, the true pair f 0 ; 0 g = f0; 0:6g is equivalent to an AR(1) process with high persistence. In this case, the method with the rst choice of estimating function, the OLS estimation with forward and backward-looking components, shows the highest power for most of the pairs tested, including the almost unit root pair f ; g = f0:99; 0:99g for small sample sizes, T =50 and 100. As we increase the sample size to T = 200, the power obtained with our method for each of the three choices of estimating function increases and reaches similar values. In Tables 4 and 5 we have two pairs with one parameter closer to the unit root than 19

20 the other. In Table 4, for the true pair f 0 ; 0 g = f0:5; 0:96g, using the autocorrelations as the estimating function in our method gives more power for most of the pairs tested when the sample size is T =50 and 100. However, its power is very poor for the almost unit root pair; and as a result, the method with the rst choice of estimating function, the OLS estimation with forward and backward-looking components, is the one able to discriminate this pair with the highest power. As before, once we increase the sample size, power reaches similar values when our method is applied with the three estimating functions. In Table 5, for the true pair f 0 ; 0 g = f0:96; 0:5g, the second choice of the estimating function, the OLS estimation of the long-ar, shows the highest power for most of the pairs tested when the sample size is T =50 and 100. Still, choosing the autocorrelations as the estimating function enables to discriminate between the true pair and the almost unit root pair, since power is poor when we use the other two estimating functions, for this particular pair. When the sample size is T =200, power increases for the three estimating functions, yet we need the autocorrelations to ensure we can discriminate the almost unit root pair. Lastly, in Table 6 we examine the almost unit root pair as the true pair, f 0 ; 0 g = f0:99; 0:99g. For most of the pairs tested, applying the method with the OLS estimation of the long-ar gives the highest power when the sample size is T =50 and 100. However, for these sample sizes, using autocorrelations gives the highest power for the last three pairs tested, for which the value of the MA parameter is closer to the boundary than the value of the AR parameter. As we increase the sample size, power reaches similar values for our method with the three estimating functions. 20

21 5.1 Robustness checks In addition to our previous experiments, we conducted simple robustness checks for our method. Two examples are shown in Tables 7 and 8. We present the power values for two ARMA processes which are misspeci ed as ARMA(1,1) with a sample size T =100. In Table 7 we have an ARMA(2,1) with f 0 ; 1;0 ; 2;0 g = f0:3; 0:6; 0:85g, so we have a high persistence AR(2) mixed with a low persistence MA(1). In general, the method applied with autocorrelations o ers more power rejecting the pairs where the AR parameter is closer to the boundary than the MA parameter, while with the OLS estimation with forward and backward-looking components we obtain more power rejecting the pairs where the MA parameter is closer to the boundary than the AR parameter. The ARMA (2,1) in Table 8 with f 1;0 ; 2;0 ; 0 g = f0:3; 0:85; 0:6g, shows a high persistence AR(1) mixed with a MA(2) which has higher persistence on its second parameter. In this case, we nd that the method applied with both, the OLS estimation of forward and backward-looking components and the OLS estimation of the long-ar, is more powerful rejecting the misspeci cation than with the autocorrelations. Note that in both examples, our three choices of estimating functions provide good power, rejecting the misspeci cation for all the pairs tested. 5.2 Impulse-response con dence bands To illustrate the transformation via the computation of impulse-response con dence bands we conduct further simulation exercises. We simulate one ARMA(1,1) sample path selecting di erent combinations of the parameters f 0 ; 0 g. The setting of our model is the same we followed on the above mentioned experiments: the total number of auxiliary parameters is set to 8 on the three choices for the estimating function, H = 3 and L =

22 We conduct a grid search on the four quadrants with a step of For brevity, we show two simulations with the pairs f 0 ; 0 g = f0:99; 0:99g and f 0 ; 0 g = f0:96; 0:5g, for samples of T =50 observations, in Figures 1 and 2. There are four diagrams on each gure: the simulated path is plotted on the upper left section, the joint 95% con dence set appears on the upper right, the joint 95% con dence surface is on the lower left, and the impulse-response con dence bands are shown on the lower right section. Figure 1 and 2 present the outcomes of our method selecting the OLS estimation with forward and backward-looking components (FBLC) and the autocorrelations, respectively as estimating functions. Note that even with this small sample size, the true values of the parameters are covered in the joint 95% con dence sets and the impulse-response corresponding to the true pair is covered by our impulse-response con dence bands. However, for the almost unit root pair in Figure 1, the impulse-response function obtained from the MLE is outside the con dence bands. As expected, inverting the asymptotic Wald test based on MLE estimates is almost infeasible at corner solutions with parameters close to the unit boundary, since in many instances the variance-covariance matrix is not invertible. Comparing our impulse-response con dence bands with the impulse-response function obtained with the MLE point estimates, we conclude that the MLE can lead to spurious results when identi cation problems exists. 6 Conclusions In this paper we revisit Indirect Inference jointly with the MC test method from a con dence set two-stage simulation-based outlook focusing on ARMA processes in nite samples. Despite the simplicity of ARMA models in time series, identi cation and bound- 22

23 ary issues raise enduring complications for estimation and inference. We propose estimating equations via two sided regressions with forward and backward-looking components, backward looking autoregressions and autocorrelations. We treat the resulting objective functions as test statistics which we invert through a two-stage simulation to obtain a joint con dence set for the parameters: rst, we obtain calibrated estimates applying the previously mentioned estimating functions, then we compute the p-values applying MC test methods. Invariance principles are proposed to ensure nite sample exactness in general nuisance parameter dependent settings. The proposed impulse-responses con dence bands represent an important and empirically relevant transformation of the joint con dence set. Supportive simulation results con rm the following. In contrast to MLE-based tests, our proposed methods achieve size control and good power, irrespective of how close the parameters are to the unit root boundary. The results obtained in term of power for processes with Student-t errors are in line with the ones obtained with Gaussian errors. Our method eradicates pile up problems even with very small samples, for example, with no more than 50 observations. The signi cance level is a ected neither by at segments in the objective function, nor by how close the estimated parameters are to the unit root boundary. Finally, our studies on impulse-response con dence bands shows that relying on MLE when the processes are highly persistent can lead researches to underestimate shock e ects when conducting estimation and forecasting. 23

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29 Table 1: Empirical Rejections: True value 0 = 0:6 SIZE Gaussian MA Student-t (df=5) MA T OLS-long AR Simpli ed MLE OLS-long AR Simpli ed MLE POWER T=50 Gaussian MA Student-t (df=5) MA Tested value OLS-long AR Simpli ed OLS-long AR Simpli ed {0.00} {0.30} {0.85} {0.90} {0.96} {0.99} T=100 Gaussian MA Student-t (df=5) MA Tested value OLS-long AR Simpli ed OLS-long AR Simpli ed {0.00} {0.30} {0.85} {0.90} {0.96} {0.99} T=200 Gaussian MA Student-t (df=5) MA Tested value OLS-long AR Simpli ed OLS-long AR Simpli ed {0.00} {0.30} {0.85} {0.90} {0.96} {0.99}

30 Table 2: Empirical Rejections: True value 0 = 0:99 SIZE Gaussian MA Student-t (df=5) MA T OLS-long AR Simpli ed MLE OLS-long AR Simpli ed MLE POWER T=50 Gaussian MA Student-t (df=5) MA Tested value OLS-long AR Simpli ed OLS-long AR Simpli ed {0.00} {0.30} {0.60} {0.85} {0.90} {0.96} T=100 Gaussian MA Student-t (df=5) MA Tested value OLS-long AR Simpli ed OLS-long AR Simpli ed {0.00} {0.30} {0.60} {0.85} {0.90} {0.96} T=200 Gaussian MA Student-t (df=5) MA Tested value OLS-long AR Simpli ed OLS-long AR Simpli ed {0.00} {0.30} {0.60} {0.85} {0.90} {0.96}

31 Table 3: Empirical Rejections: True pair f 0 ; 0 g = f0; 0:6g SIZE Gaussian ARMA Student-t (df=5) ARMA T OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr MLE Gaussian ARMA Student-t (df=5) ARMA T OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr POWER T=50 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99}

32 POWER f 0 ; 0 g = f0; 0:6g T=100 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99} T=200 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99}

33 Table 4: Empirical Rejections: True pair f 0 ; 0 g = f0:5; 0:96g SIZE Gaussian ARMA Student-t (df=5) ARMA T OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr MLE Gaussian ARMA Student-t (df=5) ARMA T MA AR Joint MA AR Joint POWER T=50 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99}

34 POWER f 0 ; 0 g = f0:5; 0:96g T=100 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99} T=200 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99}

35 Table 5: Empirical Rejections: True pair f 0 ; 0 g = f0:96; 0:5g SIZE Gaussian ARMA Student-t (df=5) ARMA T OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr MLE Gaussian ARMA Student-t (df=5) ARMA T MA AR Joint MA AR Joint POWER T=50 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.99, 0.99}

36 POWER f 0 ; 0 g = f0:96; 0:5g T=100 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.99, 0.99} T=200 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.99, 0.99}

37 Table 6: Empirical Rejections: True pair f 0 ; 0 g = f0:99; 0:99g SIZE Gaussian ARMA Student-t (df=5) ARMA T OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr MLE Gaussian ARMA Student-t (df=5) ARMA T MA AR Joint MA AR Joint POWER T=50 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5}

38 POWER f 0 ; 0 g = f0:99; 0:99g T=100 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} T=200 Gaussian ARMA Student-t (df=5) ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5}

39 Table 7: Empirical Rejections: True values f 1;0 ; 2;0 ; 0 g = f0:3; 0:85; 0:60g POWER T=100 Gaussian ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99} Table 8: Empirical Rejections: True values f 0 ; 1;0 ; 2;0 g = f0:3; 0:6; 0:85g POWER T=100 Gaussian ARMA Tested pair OLS-FBLC OLS-long AR AutoCorr {0, 0.6} {0.3, 0.2} {0.3, 0.85} {0.5, 0.5} {0.5, 0.96} {0.6, 0} {0.6, 0.85} {0.85, 0.2} {0.85, 0.6} {0.96, 0.5} {0.99, 0.99}

40 Figure 1: ARMA(1,1) Simulated Path f 0 ; 0 g = f0:99; 0:99g 40

41 Figure 2: ARMA(1,1) Simulated Path f 0 ; 0 g = f0:96; 0:5g 41